The set of monomial convergence of the bounded holomophic functions on B_{c0} and of m-homogeneous polynomials on c0 was studied in Chapter 10. Here the space c0 is replaced by some other l_p spaces, or even by polynomials on an arbitrary Banach sequence space and holomorphic functions on Reinhardt domains. The only complete case is p=1, where the set of monomial convergence of the m-homogeneous polynomials is exactly l_1, and the set of monomial convergence of the bounded holomorphic functions on the open unit ball of l_1 is again the ball. For other p’s upper and lower bounds are presented that give a pretty tight description.

Given a family of formal power series, its set of monomial convergence is defined as those z’s for which the series converges. The main focus is given to the sets of monomial convergence of the m-homogeneous polynomials on c0 and of the bounded holomorphic functions on B_{c0}. The first one is completely described in terms of the Marcinkiewicz space l_{(2m)/(m-1), ∞}. For the second one there is no complete description. If z is such that limsup (log n)^(1/2) ∑_j^n (z*_j)^{2} < 1 (where z* is the decreasing rearrangement of z), then z is in the set of monomial convergence of the bounded holomorphic functions. Also, if z belongs to the set of monomial convergence, then the limit superior is ≤ 1. This is related to Bohr’s problem (see Chapter 1). First of all, if M denotes the supremum over all q so that l_q is contained in the set of monomial convergence of the bounded holomorphic functions on Bc0, then S=1/M. But this can be more precise: S is the infimum over all σ >0 so that the sequence (p_n^(-σ))_n (being p_n the n-th prime number) belongs to the set of monomial convergence of the bounded holomorphic functions on Bc0.

This is a short introduction to the theory of holomorphic functions in finitely and infinitely many variables. We begin with functions in finitely many variables, giving the definition of holomorphic function. Every such function has a monomial series expansion, where the coefficients are given by a Cauchy integral formula. Then we move to infinitely many variables, considering functions defined on B_{c0}, the open unit ball of the space of null sequences. Holomorphic functions are defined by means of Fréchet differentiability. We have versions of Weierstrass and Montel theorems in this setting. Every holomorphic function on B_{c0} defines a family of coefficients through a Cauchy integral formula and a (formal) monomial series expansion. Every bounded analytic (represented by a convergent power series) function is holomorphic. Hilbert’s criterion, that gives conditions on a family of scalars so that it is attached to a bounded holomorphic function on B_{c0}. Homogeneous polynomials are those entire functions having non-zero coefficients only for multi-indices of a given order. We show how these are related to multilinear forms on c0 through the polarization formulas.

We study the relationship between Hardy spaces of functions on the polytorus and certain spaces of holomorphic functions. We deal first with functions in finitely many variables, and later we jump to the infinite dimensional setting. For each N we consider the space of holomorphic functions g on the N-dimensional polydisc for which the L_p norms of g(rz) for 0<r<1 are bounded (known as the Hardy space of holomorphic functions). For each p these two Hardy spaces (of integrable functions on the N-dimensional polytorus and the N-dimensional polydisc) are isometrically isomorphic. The main tool in the proof is the Poisson operator (defined in Chapter 5). For the infinite dimensional case, we define the space of holomorphic functions g on l_2 ∩ Bc0 whose restrictions to the first N variables all belong to the corresponding Hardy space, and the norms are uniformly bounded (in N). These Hardy spaces of holomorphic functions on l_2 ∩ Bc0 and the Hardy spaces of integrable functions on the infinite-dimensional polytorus are isometrically isomorphic. The jump is given using a Hilbert criterion for Hardy spaces.

We establish a bijection between Dirichlet series and formal power series through Bohr’s transform. This is one of the main tools all along the text and relies on the fact that by the fundamental theorem of arithmetic every natural number has a unique decomposition as a product of prime numbers. In this way, to each such number a multi-index can be assigned (and vice-versa). With this we show that the space of bounded holomorphic functions on B_{c0} and \mathcal{H}_\infty are isomorphic as Banach spaces. This means that to every holomorphic function corresponds a Dirichlet series in such a way that the monomial and the Dirichlet coefficients are identified. We consider m-homogenous Dirichlet series: those having non-zero coefficients only if n has exactly m prime divisors (counted with multiplicity) and show that the space of such Dirichlet series is isometrically isomorphic to the space of m-homogeneous polynomials on c0.

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.